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- Newsgroups: sci.math
- Path: sparky!uunet!morrow.stanford.edu!CSD-NewsHost.Stanford.EDU!Sunburn.Stanford.EDU!pratt
- From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
- Subject: Re: Zeno
- Message-ID: <1992Oct8.172550.2253@CSD-NewsHost.Stanford.EDU>
- Sender: news@CSD-NewsHost.Stanford.EDU
- Organization: Computer Science Department, Stanford University.
- References: <1992Oct8.000340.1@opie.bgsu.edu>
- Date: Thu, 8 Oct 1992 17:25:50 GMT
- Lines: 49
-
- In article <1992Oct8.000340.1@opie.bgsu.edu> bc205cs@opie.bgsu.edu writes:
- >Now here is my question: The rule of the game is that you can only step half
- >of the remaining distance on each step. Well, if you are standing at Point B,
- >that implies that you stepped onto point B. If you stepped _onto_ point B,
- >then you broke the rule! You were only supposed to step _halfway_ to point B.
- >Therefore, Zeno's paradox holds, and the greatest minds are wrong.
-
- I like this paradox. It would appear to among the oldest surviving
- instances of common sense conflicting with the logic of infinities and
- infinitesimals. As they say, something's gotta give.
-
- To put the paradox in its best light, consider yourself walking slowly
- but steadily the mile from A to B at one mile per hour. Clearly you
- reach B one hour after leaving A. Yet your motion can also be
- described as "stepping" halfway from A to B in half an hour, thence
- half the remaining distance in quarter of an hour, and so on "for
- ever." At which of those steps did you reach B? Surely none, so you
- can never reach B.
-
- Here's one way out of the paradox. Implicit in your argument is that
- you stepped onto point B from a preceding point. But who says every
- step has to have a predecessor? Certainly the 1st step steps from A,
- and so on with the 2nd and 3rd steps. But how does that prove that
- *every* step steps from a particular point? In fact we can turn the
- argument in the paradox around to make it into a simple proof of the
- claim that *not every step is from a preceding point*, in particular
- the step that lands you on B.
-
- So what is the number of the step that lands on B? Not 666, or 10!!,
- or any finite integer. Rather it is the first infinite integer, better
- known as the smallest limit ordinal, namely omega. If we include for
- completeness a fictitious "step 0" landing us at A, then we have two
- steps neither having a predecessor, unlike all the steps in between.
- Of the two predecessor-less steps, step 0 was not a limit step, not
- being the limit of anything, but step omega certainly was the limit of
- the process of stepping towards B. Omega is the first limit ordinal.
-
- Future research. Now that we've reconciled Zeno's two pictures of
- travel from A to B, at least for those willing to accept the idea of a
- step with an infinite number, how does each picture extend to travel on
- beyond B? The steady-motion picture is fine, but what about the
- ordinal picture? Do we go to higher ordinals, or what? Or have we
- simply stepped out of the frying pan into the fire by replacing one
- paradox with another? Left to the reader.
-
- --
- =================================================== In short, I am floundering
- Vaughan Pratt pratt@cs.Stanford.EDU 415-494-2545 around and don't take what
- =================================================== I say seriously.
-